On Mixing Distributions Via Random Orthogonal Matrices and the Spectrum of the Singular Values of Multi-Z Shaped Graph Matrices

Abstract

In this paper, we introduce and analyze a new operation • R which mixes two distributions Ω and Ω via a random orthogonal matrix. In particular, we take Ω • R Ω to be the limit as n → ∞ of the distribution of singular values of DRD where D and D are n × n diagonal matrices whose diagonal entries have distributions Ω and Ω respectively and R is a random n × n orthogonal matrix.We show that • R has several nice properties. We first observe that • R is commutative and associative and compute the moments of Ω • R Ω in terms of the moments of Ω and Ω . We then show that • R interacts very nicely with the spectrum of the singular values of Z-shaped and multi-Z-shaped graph matrices. This allows us to answer the question posed by our previous paper of how to describe the spectrum of the singular values of Z-shaped and multi-Z-shaped graph matrices when the input distribution is not {−1, 1}.In our analysis, we show that the moments of our distributions are closely connected to non-crossing partitions and prove a number of new results on non-crossing partitions which may be of independent interest.